Groups

Abbreviation: Grp

Definition

A group is a structure G = (G,·,⁻¹,e), where · is an infix binary operation, called the group product, ⁻¹ is a postfix unary operation, called the group inverse and e is a constant (nullary operation), called the identity element, such that
· is associative: (xy)z = x(yz),
e is a left-identity for ·: ex = x, and
⁻¹ gives a left-inverse: x⁻¹x = e.

Remark: It follows that e is a right-identity and that ⁻¹gives a right inverse: xe = x, xx⁻¹ = e.
This definition shows that groups form a variety.

Morphisms

Let G and H be groups. A morphism from G to H is a function h : G→H that is a homomorphism: h(xy) = h(x)h(y) and h(x⁻¹) = h(x)⁻¹ and h(e) = e.

Some results

Examples

(S_X,o,⁻¹,id_X), the collection of permutations of a sets X, with composition, inverse, and identity map.

The general linear group (GL_n(V),·,⁻¹,I_n), the collection of invertible n×n matrices over a vector space V, with matrix multiplication, inverse, and identity matrix.

Properties

Classtype	variety
Equational theory	decidable in polynomial time
Quasiequational theory	undecidable
First-order theory	undecidable
Congruence distributive	no (Z₂×Z₂)
Congruence modular	yes
Congruence n-permutable	yes, n=2, p(x,y,z) = xy⁻¹z is a Mal'cev term
Congruence regular	yes
Congruence uniform	yes
Congruence types	1=permutational
Congruence extension property	no, consider a non-simple subgroup of a simple group
Definable principal congruences
Equationally definable principal congruences	no
Amalgamation property	yes
Strong amalgamation property	yes
Epimorphisms are surjective	yes
Locally finite	no
Residual size	unbounded

Finite members

[Size 1]?: 1
[Size 2]?: 1
[Size 3]?: 1
[Size 4]?: 2
[Size 5]?: 1
[Size 6]?: 2
[Size 7]?: 1
[Size 8]?: 5
[Size 9]?: 2
[Size 10]?: 2
[Size 11]?: 1
[Size 12]?: 5
[Size 13]?: 1
[Size 14]?: 2
[Size 15]?: 1
[Size 16]?: 14
[Size 17]?: 1
[Size 18]?: 5

Information about small groups up to size 2000: http://www.tu-bs.de/~hubesche/small.html

Subclasses

p-groups?
[nilpotent groups]?
[solvable groups]?

Superclasses

Monoids
Inverse semigroups